Series HRS amob Z. Roll No. H$moS> Z. 0/ (SPL) Code No. narjmwu H$moS >H$mo CÎma-nwpñVH$m Ho$ _wi-n ð >na Adí` {bio & Candidates must write the Code on the title page of the answer-book. H $n`m Om±M H$a b {H$ Bg àíz-nì _o _w{ðv n ð> 5 h & àíz-nì _ Xm{hZo hmw H$s Amoa {XE JE H$moS >Zå~a H$mo N>mÌ CÎma-nwpñVH$m Ho$ _wi-n ð> na {bi & H $n`m Om±M H$a b {H$ Bg àíz-nì _ >4 àíz h & H $n`m àíz H$m CÎma {bizm ewê$ H$aZo go nhbo, àíz H$m H«$_m H$ Adí` {bi & Bg àíz-nì H$mo n T>Zo Ho$ {be 5 {_ZQ >H$m g_` {X`m J`m h & àíz-nì H$m {dvau nydm _ 0.5 ~Oo {H$`m OmEJm & 0.5 ~Oo go 0.0 ~Oo VH$ N>mÌ Ho$db àíz-nì H$mo n T> Jo Am a Bg Ad{Y Ho$ Xm amz do CÎma-nwpñVH$m na H$moB CÎma Zht {bi Jo & Please check that this question paper contains 5 printed pages. Code number given on the right hand side of the question paper should be written on the title page of the answer-book by the candidate. Please check that this question paper contains 4 questions. Please write down the Serial Number of the question before attempting it. 5 minutes time has been allotted to read this question paper. The question paper will be distributed at 0.5 a.m. From 0.5 a.m. to 0.0 a.m., the students will read the question paper only and will not write any answer on the answer-book during this period. g H${bV narjm II SUMMATIVE ASSESSMENT II J{UV MATHEMATICS {ZYm [av g_` : KÊQ>o A{YH$V_ A H$ : 90 Time allowed : hours Maximum Marks : 90 0/ (SPL) P.T.O.
gm_mý` {ZX}e : (i) g^r àíz A{Zdm` h & (ii) Bg àíz-nì _ 4 àíz h Omo Mma IÊS>m A, ~, g Am a X _ {d^m{ov h & (iii) IÊS> A _ EH$-EH$ A H$ dmbo 8 àíz h, Omo ~hþ-{dh$ënr àíz h & IÊS> ~ _ 6 àíz h {OZ_ go àë`oh$ 2 A H$ H$m h & IÊS> g _ 0 àíz VrZ-VrZ A H$m Ho$ h & IÊS> X _ 0 àíz h {OZ_ go àë`oh$ 4 A H$ H$m h & (iv) H $bhw$boq>a H$m à`moj d{o V h & General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 4 questions divided into four sections A, B, C and D. (iii) (iv) Section A contains 8 questions of mark each, which are multiple choice type questions, Section B contains 6 questions of 2 marks each, Section C contains 0 questions of marks each and Section D contains 0 questions of 4 marks each. Use of calculators is not permitted. IÊS> A SECTION A àíz g»`m go 8 VH$ àë`oh$ àíz A H$ H$m h & àíz g»`m go 8 _ àë`oh$ àíz Ho$ {be Mma {dh$ën {XE JE h, {OZ_ go Ho$db EH$ ghr h & ghr {dh$ën Mw{ZE & Question numbers to 8 carry mark each. For each of the question numbers to 8, four alternative choices have been provided, of which only one is correct. Select the correct choice.. PQ EH$ d Îm na {~ÝXþ P go JwµOaZo dmbr ñne aoim h {OgHo$ d Îm H$m Ho$ÝÐ-{~ÝXþ O h & `{X OPQ EH$ g_{û~mhþ {Ì^wO h, Vmo OQP H$m _mz h 0 45 60 90 PQ is a tangent to a circle with centre O at the point P. If OPQ is an isosceles triangle, then OQP is equal to 0 45 60 90 0/ (SPL) 2
2. AmH ${V _, RQ d Îm na EH$ ñne aoim h {OgH$m Ho$ÝÐ-{~ÝXþ O h & `{X SQ = 6 go_r VWm QR = 4 go_r h, Vmo OR H$s b ~mb h 8 go_r go_r 2. 5 go_r 5 go_r AmH ${V In figure, RQ is a tangent to the circle with centre O. If SQ = 6 cm and QR = 4 cm, then OR is Figure 8 cm cm 2. 5 cm 5 cm 0/ (SPL) P.T.O.
. EH$ nv J VWm ^y{_ na EH$ {~ÝXþ Ho$ ~rm ~±Yr S>moar H$s b ~mb 85 _r. h & `{X S>moar ^y{_vb Ho$ gmw H$moU Bg àh$ma ~Zm ahr h {H$ tan = 5 h, Vmo nv J H$s ^y{_ go 8 D±$MmB {H$VZr h? 75 _r. 79. 4 _r. 80 _r. 72. 5 _r. The length of a string between a kite and a point on the ground is 85 m. If the string makes an angle with the ground level such that 5 tan =, then the kite is at what height from the ground? 8 75 m 79. 4 m 80 m 72. 5 m 4. `{X EH$ d Îm H$s n[a{y 8 h, Vmo CgH$m joì\$b h 8 6 4 2 If the circumference of a circle is 8, then its area is 8 6 4 2 0/ (SPL) 4
5. EH$ AÀN>r Vah \ $Q>r JB Vme Ho$ nîmm H$s JÈ>r _ go BªQ> Ho$ ~mxemh VWm ~oj_ H$mo hq>m {X`m OmVm h & {\$a ~Mo hþe nîmm _ go EH$ nîmm `mñàn>`m N>m±Q>m OmVm h & Vmo {M S>r Ho$ ~mxemh H$mo àmá H$aZo H$s àm{`h$vm h 4 4 50 52 The king and queen of diamonds are removed from a pack of well-shuffled playing cards. One card is selected at random from the remaining cards. The probability of getting the king of clubs is 4 4 50 52 6. `{X 7, x, y, 5 EH$ g_m Va lo T>r _ h, Vmo x y H$m _mz h 2 4 4 0/ (SPL) 5 P.T.O.
If 7, x, y, 5 are in A.P., then the value of x y is 2 4 4 7. `{X {~ÝXþ P(5, y), {~ÝXþAm A(, 5) VWm B(x, ) H$mo {_bmzo dmbo aoimiês> AB H$m _Ü`-{~ÝXþ h, Vmo (x + y) H$m _mz h 7 4 If P(5, y) is the mid-point of the line segment AB joining the points A(, 5) and B(x, ), then (x + y) equals 7 4 8. EH$ nmgo H$mo EH$ ~ma CN>mbZo na, 4 go ~ S>r g»`m AmZo H$s àm{`h$vm h 5 6 6 2 0/ (SPL) 6
The probability of getting a number greater than 4, when a die is rolled once, is 5 6 6 2 IÊS> ~ SECTION B àíz g»`m 9 go 4 VH$ àë`oh$ àíz Ho$ 2 A H$ h & Question numbers 9 to 4 carry 2 marks each. 9. PA VWm PB EH$ ~mø {~ÝXþ P go O Ho$ÝÐ dmbo d Îm H$s ñne aoime± h, Omo d Îm H$mo H«$_e: {~ÝXþAm A VWm B na ñne H$aVr h & Xem BE {H$ MVw^w O AOBP MH«$s` MVw^w O h & PA and PB are tangents to the circle with centre O from an external point P, touching the circle at A and B respectively. Show that the quadrilateral AOBP is cyclic. 0. EH$ {n½jr ~ H$ _ 50 n go Ho$ 00 {g o$, < Ho$ 50 {g o$, < 2 Ho$ 20 {g o$ VWm < 5 Ho$ 0 {g o$ h & `{X Bg ~ H$ H$mo CbQ>m H$aZo na àë`oh$ {g o$ Ho$ ~mha {JaZo H$s g ^mdzm ~am~a h, Vmo Š`m àm{`h$vm hmojr {H$ {JaZo dmbm nhbm {g $m < 5 H$m {g $m Zht h? A piggy bank contains hundred 50 paise coins, fifty < coins, twenty < 2 coins and ten < 5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, what is the probability that the first coin will not be a < 5 coin? 0/ (SPL) 7 P.T.O.
. AmH ${V 2 _, ABCD EH$ Eogm MVw^w O h {Og_ D = 90 h & EH$ d Îm {OgH$m Ho$ÝÐ O VWm {ÌÁ`m r h, MVw^ wo H$s ^womam AB, BC, CD VWm DA H$mo H«$_e: P, Q, R VWm S na ñne H$aVm h & `{X BC = 40 go_r, CD = 25 go_r VWm BP = 28 go_r h, Vmo r H$m _mz kmv H$s{OE & AmH ${V 2 In figure 2, ABCD is a quadrilateral such that D = 90. A circle with centre O and radius r, touches the sides AB, BC, CD and DA at P, Q, R and S respectively. If BC = 40 cm, CD = 25 cm and BP = 28 cm, find r. Figure 2 2. grgo go ~Zo hþe EH$ R>mog KZ, {OgH$s ^wom H$s _mn 44 go_r h, _ go {H$VZr JmobmH$ma R>mog Jmo{b`m± ~ZmB Om gh$vr h, `{X àë`oh$ Jmobr H$m ì`mg 4 go_r h & [ = 7 22 br{oe ] How many spherical solid bullets can be made out of a solid cube of lead whose edge measures 44 cm, each bullet being 4 cm in diameter. [ Use = 7 22 ] 0/ (SPL) 8
. EH$ g_m Va lo T>r H$m Vrgam Ed 7dm± nx H«$_e VWm h, Vmo BgH$m ndm± nx kmv H$s{OE & The third and the 7 th terms of an A.P. are and respectively. Find the n th term of the A.P. 4. k Ho$ {H$g _mz Ho$ {be {ÛKmV g_rh$au kx 2 2 (k + 2) x + (k + 5) = 0 Ho$ _yb dmñv{dh$ VWm g_mz hm Jo? For what value of k are the roots of the quadratic equation kx 2 2 (k + 2) x + (k + 5) = 0 real and equal? IÊS> g SECTION C àíz g»`m 5 go 24 VH$ àë`oh$ àíz Ho$ A H$ h & Question numbers 5 to 24 carry marks each. 5. AmH ${V _, Xmo g H $Ðr` d Îmmo H$s {ÌÁ`mE± go_r VWm 8 go_r h & AB ~ S>r {ÌÁ`m dmbo d Îm H$m ì`mg h VWm BD N>moQ>o d Îm H$s$ {~ÝXþ D na EH$ ñne aoim h & AD H$s b ~mb kmv H$s{OE & AmH ${V 0/ (SPL) 9 P.T.O.
In figure, the radii of two concentric circles are cm and 8 cm. AB is a diameter of the bigger circle and BD is a tangent to the smaller circle touching it at D. Find the length of AD. Figure 6. EH$ \$mc Q>oZ noz H$m ~ ab, ~obzmh$ma h VWm BgH$s b ~mb 7 go_r d ì`mg 5 {_br_rq>a h & `{X ñ`mhr go nyao ^ao ~ ab go, Am gvz 0 eãx {bio Om gh$vo h, Vmo {H$gr ~movb {Og_ EH$ brq>a H$m nm±mdm± ^mj ñ`mhr ^ar h, go {H$VZo eãx {bio Om gh $Jo? The barrel of a fountain-pen, cylindrical in shape, is 7 cm long and 5 mm in diameter. If with a full barrel of ink, on an average 0 words can be written, then how many words would use up a bottle of ink containing one-fifth of a litre? 7. EH$ ^dz H$m Am V[aH$ ^mj bå~-d Îmr` ~obz Ho$ ê$n _ h VWm CgH$m ì`mg 4. 2 _r. Ed D±$MmB 4 _r. h & ~obzmh$ma ^mj Ho$ D$na g_mz ì`mg H$m EH$ e Hw$ Amamo{nV h {OgH$s D±$MmB 2. 8 _r. h & ^dz H$m ~mhar n ð>r` joì\$b kmv H$s{OE & The interior of a building is in the form of a right circular cylinder of diameter 4. 2 m and height 4 m surmounted by a cone of same diameter. The height of the cone is 2. 8 m. Find the outer surface area of the building. 8. {H$gr ju EH$ ÜdOXÊS> H$s N>m`m H$s b ~mb, Cg g_` H$s BgH$s N>m`m H$s b ~mb H$s VrZ JwZr h, O~ gy` H$s {H$aU ^y{_ Ho$ gmw 60 H$m H$moU ~ZmVr h & Bg ju H$m dh H$moU kmv H$s{OE Omo gy` H$s {H$aU ^y{_ Ho$ gmw ~ZmVr h & At an instant the shadow of a flag-staff is three times as long as its shadow, when the sun-rays make an angle of 60 with the ground. Find the angle between the sun-rays and the ground at this instant. 0/ (SPL) 0
9. dh AZwnmV kmv H$s{OE {Og_ {~ÝXþ Q (, p), {~ÝXþAm A ( 5, 4) VWm B ( 2, ) H$mo {_bmzo dmbo aoimiês> AB H$mo {d^m{ov H$aVm h & p H$m _mz ^r kmv H$s{OE & Find the ratio in which the point Q (, p) divides the line segment AB joining the points A ( 5, 4) and B ( 2, ). Also, find the value of p. 20. EH$ e Hw$ Ho$ {N>ÞH$ Ho$ XmoZm d Îmr` {gam Ho$ n[a_mn 48 go_r VWm 6 go_r h & `{X {N>ÞH$ H$s D±$MmB go_r h, Vmo CgH$m Am`VZ kmv H$s{OE & [ = 7 22 br{oe ] The perimeters of the two circular ends of a frustum of a cone are 48 cm and 6 cm. If the height of the frustum is cm, find its volume. [Use = 7 22 ] 2. g_m Va lo T>r 48, 42, 6, Ho$ {H$VZo nxm H$m `moj\$b 26 h? Xmo CÎma AmZo Ho$ H$maU H$s ì`m»`m H$s{OE & How many terms of the A.P. 48, 42, 6, be taken so that the sum is 26? Explain the double answer. 22. 4 go_r {ÌÁ`m dmbo EH$ d Îm H$s EH$ Ordm, d Îm Ho$ Ho$ÝÐ na 60 H$m H$moU A V[aV H$aVr h & g JV bkw VWm XrK d ÎmIÊS>m Ho$ joì\$b kmv H$s{OE & [ = 7 22 VWm =. 7 br{oe] A chord of a circle of radius 4 cm, subtends an angle of 60 at the centre. Find the areas of the corresponding minor and major segments of the circle. [Use = 7 22 and =. 7] 2. Ma x _ {ÛKmV g_rh$au 2 abx 2 (9a 2 8b 2 ) x 6ab = 0 Ho$ _yb kmv H$s{OE & Find the roots of the quadratic equation 2 abx 2 (9a 2 8b 2 ) x 6ab = 0 in the variable x. 24. {Ì^wO PQR H$s _mpü`h$mam RS VWm PT H$s b ~mb`m± kmv H$s{OE, O~{H$ {Ì^wO Ho$ erf P(6, 2), Q(6, ) VWm R(, ) h & Find the lengths of the medians RS and PT of a triangle PQR whose vertices are P(6, 2), Q(6, ) and R(, ). 0/ (SPL) P.T.O.
IÊS> X SECTION D àíz g»`m 25 go 4 VH$ àë`oh$ àíz Ho$ 4 A H$ h & Question numbers 25 to 4 carry 4 marks each. 25. AmH ${V 4 _, N>m`m {H$V ^mj H$m joì\$b kmv H$s{OE, Ohm± 2 go_r ^wom dmbo EH$ g_~mhþ {Ì^wO OAB Ho$ erf O H$mo Ho$ÝÐ _mzh$a 6 go_r {ÌÁ`m dmbm d Îmr` Mmn ItMm J`m VWm erf B H$mo Ho$ÝÐ boh$a 6 go_r {ÌÁ`m Ho$ d Îm H$m EH$ {ÌÁ`IÊS> ~Zm`m J`m h & AmH ${V 4 Find the area of the shaded region in figure 4, where a circular arc of radius 6 cm has been drawn with vertex O of an equilateral triangle OAB of side 2 cm as centre and a sector of circle of radius 6 cm with centre B is made. Figure 4 0/ (SPL) 2
26. EH$ _moq>a-zm H$m, {OgH$s pñwa Ob _ Mmb 24 {H$_r à{v K Q>m h, 2 {H$_r Ymam Ho$ à{vhy$b OmZo _, dhr Xÿar Ymam Ho$ AZwHy$b OmZo H$s Anojm K Q>m A{YH$ bovr h & Ymam H$s Mmb kmv H$s{OE & A motorboat, whose speed is 24 km/h in still water, takes hour more to go 2 km upstream than to return downstream to the same spot. Find the speed of the stream. 27. I S>r MÅ>mZ na I S>m EH$ ì`{º$ EH$ Zmd H$mo AnZo R>rH$ ZrMo {H$Zmao H$s Amoa EH$g_mZ Mmb go 0 Ho$ AdZ_Z H$moU na AmVm XoI ahm h & 6 {_ZQ> Ho$ nímmv² Zmd H$m `h AdZ_Z H$moU 60 hmo OmVm h & Zmd Ûmam {H$Zmao na nhþ±mzo H$m Hw$b g_` kmv H$s{OE & A man on a cliff observes a boat at an angle of depression of 0 which is approaching the shore to the point immediately beneath the observer with a uniform speed. Six minutes later the angle of depression of the boat is found to be 60. Find the total time taken by the boat to reach the shore. 28. 2 H$mo Mma Eogo ^mjm _ {d^m{ov H$s{OE {H$ `h Mmam ^mj EH$ g_m Va lo T>r Ho$ Mma nx hm VWm nhbo d Mm Wo nxm Ho$ JwUZ\$b VWm Xÿgao d Vrgao nxm Ho$ JwUZ\$b _ 7 : 5 H$m AZwnmV hmo & Divide 2 into four parts which are the four terms of an A.P., such that the product of the first and the fourth terms is to the product of the second and the third terms as 7 : 5. 29. ABC {OgHo$ erf -{~ÝXþ A(0, ), B(2, ) VWm C(0, ) h, H$m joì\$b kmv H$s{OE & BgH$s ^womam Ho$ _Ü`-{~ÝXþAm H$mo {_bmh$a ~ZmE JE {Ì^wO H$m joì\$b ^r kmv H$s{OE & Xem BE {H$ XmoZm {Ì^wOm Ho$ joì\$bm _ 4 : H$m AZwnmV h & Find the area of ABC with vertices A(0, ), B(2, ) and C(0, ). Also find the area of the triangle formed by joining their mid-points. Show that the ratio of the areas of the two triangles is 4 :. 0/ (SPL) P.T.O.
0. EH$ {Ì^wO ABC H$s amzm H$s{OE {OgH$s ^wom BC = 7 go_r, B = 45 Am a A = 05 h & BgHo$ nímmv² EH$ AÝ` {Ì^wO H$s amzm H$s{OE {OgH$s ^wome± {Ì^wO ABC H$s g JV ^womam H$s 4 JwZm hm & Draw a triangle ABC with side BC = 7 cm, B = 45 and A = 05. Then construct another triangle whose sides are 4 times the corresponding sides of ABC.. EH$ µoma _ VrZ {^Þ a Jm Zrbo, hao VWm gµ\o$x a J Ho$ Hw$b 54 H $Mo h & EH$ Zrbo a J Ho$ H $Mo H$mo `mñàn>`m {ZH$mbZo H$s àm{`h$vm VWm hao a J Ho$ H $Mo H$mo$ `mñàn>`m {ZH$mbZo H$s àm{`h$vm 9 4 h & Bg µoma _ {H$VZo gµ\o$x a J Ho$ H $Mo h? A jar contains 54 marbles of three different colours blue, green and white. The probability of drawing a blue marble at random is and that of a green marble is 9 4. How many white marbles are there in the jar? 2. 50 H$m_Jmam H$mo EH$ H$m` H$mo Hw$N> {XZm _ g_má H$aZo Ho$ {be {Z`wº$ {H$`m J`m & 4 H$m_Jmam Zo Xÿgao {XZ H$m` N>mo S> {X`m & 4 Am a H$m_Jmam Zo Vrgao {XZ H$m` N>mo S> {X`m & Bgr àh$ma H«$_e ha {XZ hmovm J`m & Eogm H$aZo go H$m` {ZYm [av g_` go 8 A{YH$ {XZm _ g_má hþam & Vmo H$m` Hw$b {H$VZo {XZm _ nyam hþam? 50 workers were engaged to finish a piece of work in a certain number of days. Four workers dropped the second day, four more workers dropped the third day and so on. It took 8 more days to finish the work. Then find the number of days in which the work was completed. 0/ (SPL) 4
. 4 _r. ì`mg H$m EH$ Hw$Am± 4 _r. JhamB VH$ ImoXm OmVm h Am a ImoXZo go {ZH$br hþb {_Å>r H$mo Hw$E± Ho$ Mmam Amoa 4 _r. Mm S>r EH$ d ÎmmH$ma db` (ring) ~ZmVo hþe, g_mz ê$n go \ $bmh$a EH$ àh$ma H$m ~m±y ~Zm`m OmVm h & Bg ~m±y H$s D±$MmB kmv H$s{OE & `h àíz {H$g _yë` H$mo Xem Vm h? A well of diameter 4 m is dug 4 m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 4 m to form an embankment. Find the height of the embankment. What value is shown in this question? 4. {gõ H$s{OE {H$ d Îm Ho$ {H$gr {~ÝXþ na ñne aoim, ñne {~ÝXþ go hmoh$a OmZo dmbr {ÌÁ`m na b ~ hmovr h & Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. 0/ (SPL) 5 P.T.O.